Carlos Kleiber, for many the greatest conductor of all time, is an enigma. Charles Barber's new book gives us the troubled, funny perfectionist behind the ecstatic music-making

There are musical myths – and then there's Carlos Kleiber. The conductor – voted last year by 100 members of his profession as the greatest of all time, ever, in BBC Music Magazine – was, even before his death in 2004, the embodiment of the enigmatic reclusive genius – the maestro who, as Herbert von Karajan put it, would only conduct when his freezer was empty. A thumbnail of the Kleiber myths goes something like this: he was the "perfect conducting machine", in Gunther Schuller's words, who hardly ever conducted; he was a musical genius who knew the entire orchestral and operatic repertoire but only had a tiny selection of pieces he ever played in public; he was one of the funniest, most communicative musicians who ever lived, but never gave an interview; he was tormented by the ghost of his father, the great conductor Erich Kleiber; and he once gave a concert as long as his fee was a new Audi A8 with all the trimmings.

There are grains of truth in all of those (the Audi one is definitely true), but there's much, much more to Kleiber than the myth-making. At least there is now, thanks to Charles Barber's astonishing new book, Corresponding with Carlos: A Biography of Carlos Kleiber. Charles had a unique relationship with Kleiber. As a conducting student at Stanford University, with dazzling boldness and naivety, he wrote to Kleiber out of the blue and said he wanted to study with him. The key was Barber's use of humour and irony to attempt to elicit a response from Kleiber – it worked. Barber never formally became a student of Kleiber's (nobody ever did), but from 1989 until the maestro's death, he corresponded with the supposedly unknowable Carlos, and as well as vivid account of Kleiber's life, Barber's book publishes pretty well the complete letters he received.

And they're a revelation. Kleiber proves as virtuosically funny and self-deprecating as he was incandescent on the podium. "The bottom line always seems to be: no one on earth can tell you anything accurate or intelligent about conductors, least of all musicians, critics, and … CONDUCTORS, including yours sincerely. Why? Because all and sundry don't have the faintest, including, again, me." The bulk of the exchange centres on the films of the great maestros of yesteryear that Barber was collating for his Conductors on Film collection, which he sent – 51 video tapes in all – to Kleiber. Carlos's responses are fascinating, his letters executing pirouettes of musical and literary meaning in nearly every sentence, and each disproving his own maxim that he can't say anything meaningful about his life's work. On how listening to Duke Ellington gave him the clue about how to conduct Beethoven's Coriolan Overture: "The Duke and I whipped a downbeat sans upbeat out of nowhere for the start and similar 'starts', making it sound like running into a wall at 60mph with a Rolls-Royce, OK?" On conducting from memory: "'Doing' a piece 'from memory' is something your Aunt Sally would have no trouble with. Knowing exactly what is (supposed to be) going on is something, I believe, only [Dimitri] Mitropoulos could honestly claim to. With the right band in a good, condescending mood, there'd be no audible difference between Sally and Dimitri, if Sally had digested the overall ductus." There's also real insight into his way of thinking about music's relationship with the world. He loved Emily Dickinson, and often quotes her poetry in the letters, saying he was the reincarnation of her dog, Carlo, and (quixotically, but passionately) hated Abraham Lincoln – something Barber tries and fails to change his mind on.

Above all, Kleiber worked. Hard. The clue to the supposedly mystical power of Kleiber's conducting proves not so elusive after all: as Barber's book shows, he worked more fastidiously and more intensely when he did conduct than any other musician, studying the manuscript, where possible, of every piece he played, and listening to every performance he could gets his hands on.

Barber's book does more than any other I know to simultaneously reveal the truth behind the Kleiber myths and to illuminate the deeper mystery of how his recordings and films continue to have such a talismanic power. This is a brilliant summary of Kleiber's way of making music, I think: "When he heard a piece in his mind, he saw each phrase in all its iterations moving nearer to the originating code of conception – perhaps just a single note. His rehearsals operated the same way, always moving toward an infinite point of truth just over there, just past the visible horizon. And he worked in the opposite direction, simultaneously … When he beat the first bar of a great work, in his mind he was already in the last."

Still not a Kleiber convert? Then do as Charles says, and start by watching Kleiber conduct a Strauss waltz from one of the two New Year's Day concerts he conducted with the Vienna Philharmonic, Die Libelle, which is a miniature masterclass in, well, virtually all you need to know about great conducting; about the poetic, alchemical connection between gesture and sound. And then watch his Beethoven symphonies, the Fourth and the Seventh, with the Concertgebouw – and you'll never look back. For the serious Kleiberophile, YouTube has some truly amazing things: volodya2 has uploaded rare rehearsal footage of him conducting Strauss's Rosenkavalier in Vienna in 1994, and Wagner's Tristan in Stuttgart in 1970; and the conductor who never gave an interview gives an interview here! The only one he ever did do, admittedly, but there it is, in German, in all its five minutes of glory. Even better, here is the complete film of the terrifyingly powerful performance of Verdi's Otello with Domingo at La Scala in 1976; and for good measure here's my personal favourite at the moment: his frighteningly moving Brahms's Second Symphony with the Vienna Philharmonic. Barber talks about Kleiber as a perfectionist and an ecstatic. If the perfectionism was his curse – the impossibly high standards that he felt he never truly reached – it's the ecstasy that we're left with. As Barber says: "I glory in the fact that he was able to make his miracles at all. Lucky him. Lucky, lucky us."